I listened to Turn Coat recently and noticed some details I think would be helpful for the timing. The heat mentioned in the first chapter notwithstanding, Harry says that the lilacs are blooming outside his apartment building-- that generally happens in May/June. Murphy is also wearing a jacket when she meets up with Harry after the encounter with the skinwalker at Billy's, which means that the evening temperatures are cool-- you don't generally need a jacket at any time of the day or night in full-on summer in Chicago. (Or rather, you don't really need it in Minneapolis, and I can't imagine that Chicago is cooler, even with the lake.)
As for locating the series relative to real-world year, Circuit City is apparently still in business, as Harry mentions that he keeps getting flyers from them in his mail. CC closed all of their physical locations in ~2009 (it was announced in January and took effect by the end of March.) Those flyers might have been for their online business, however, which apparently continued for a while, so it's not as firm a date as I would like.